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Quasiperiodic systems serve as fertile ground for studying localisation, dueto their propensity already in one dimension to exhibit rich phase diagramswith mobility edges. The deterministic and strongly-correlated nature of thequasiperiodic potential nevertheless offers challenges distinct from disorderedsystems. Motivated by this, we present a theory of localisation inquasiperiodic chains with nearest-neighbour hoppings, based on the convergenceof local propagators; exploiting the fact that the imaginary part of theassociated self-energy acts as a probabilistic order parameter for localisationtransitions and, importantly, admits a continued-fraction representation.Analysing the convergence of these continued fractions, localisation or itsabsence can be determined, yielding in turn the critical points and mobilityedges. Interestingly, we find anomalous scalings of the order parameter withsystem size at the critical points, consistent with the fractal character ofcritical eigenstates. Self-consistent theories at high orders are alsoconsidered, shown to be conceptually connected to the theory based on continuedfractions, and found in practice to converge to the same result. Results areexemplified by analysing the theory for three quasiperiodic models covering arange of behaviour.

Localisation in quasiperiodic chains: a theory based on convergence of local propagators